A condition that implies full homotopical complexity of orbits for surface

doi:10.1017/etds.2019.62 c Cambridge University Press, 2019

∗Provisional—final page numbers to be inserted when paper edition is published

A condition that implies full homotopical complexity of orbits for surface

homeomorphisms

SALVADOR ADDAS-ZANATA† and BRUNO DE PAULA JACOIA‡

†Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Brazil (e-mail: [email protected])

‡Rua do Mat˜ao 1010, Cidade Universit´aria, 05508-090 S˜ao Paulo, SP, Brazil (e-mail: [email protected])

(Received15January2019and accepted in revised form24July2019)

Abstract. We consider closed orientable surfacesSof genusg>1 and homeomorphisms f :S→Sisotopic to the identity. A set of hypotheses is presented, called a fully essential system of curvesC and it is shown that under these hypotheses, the natural lift of f to the universal cover ofS (the Poincar´e diskD), denoted by ef,has complicated and rich dynamics. In this context, we generalize results that hold for homeomorphisms of the torus isotopic to the identity when their rotation sets contain zero in the interior. In particular, forC¹⁺ diffeomorphisms, we show the existence of rotational horseshoes having non- trivial displacements in every homotopical direction. As a consequence, we found that the homological rotation set of such an f is a compact convex subset ofR^2g with maximal dimension and all points in its interior are realized by compact f-invariant sets and by periodic orbits in the rational case. Also, f has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. This implies, in case where f is area preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.

Key words: low dimensional dynamics, topological dynamics, rotation sets, rotational horseshoes, pseudo-Anosov maps

2010 Mathematics Subject Classification: 37E30, 37E45 (Primary); 37D25, 37C25 (Secondary)

1. Introduction

1.1. Preliminaries. The main motivation for this work is to generalize some results that hold for homeomorphisms and diffeomorphisms of the torus isotopic to the identity to

ρ(ef)

i≥1n≥i n ep R .

This set is a compact convex subset ofR² (see [24]), and it was proved in [10,25]

that all points in its interior are realized by compact f-invariant subsets ofT², which can be chosen as periodic orbits in the rational case. By saying that some vectorv∈ρ(ef)is realized by a compact f-invariant set, we mean that there exists a compact f-invariant subset K⊂T²such that, for all p∈K and anyep∈π⁻¹(p),whereπ:R²→T²is the associated covering map,

n→∞lim

efⁿ(ep)− ep

n =v. (2)

Moreover, the above limit, whenever it exists, is called the rotation vector of the point p, denotedρ(p).

Before presenting the results in the torus that we want to generalize to other surfaces, we need a definition.

Definition.(Topologically transverse intersections) IfMis a surface, f :M→Mis aC¹ diffeomorphism and p,q∈M are f-periodic saddle points, then we say thatW^u(p)has a topologically transverse intersection withW^s(q)(and writeW^u(p)tW^s(q)), whenever there exists a pointr∈W^s(q)∩W^u(p)(r, clearly, can be chosen arbitrarily close toq or top) and an open ballBcentered atrsuch thatB\α=B₁∪B₂, whereαis the connected component ofW^s(q)∩B which containsr and has the following property. There exists a closed connected arcβ⊂W^u(p)such thatβ⊂B, r∈β andβ\r has two connected components, one contained in B1∪αand the other contained in B2∪α, such thatβ∩ B₁6= ∅andβ∩B₂6= ∅. Clearly, aC¹-transverse intersection is topologically transverse.

Note that asβ∩αmay contain a connected arc containingr, the ballBmay not be chosen arbitrarily small.

Remark. The consequence of a topologically transverse intersection which is more relevant to us is aC⁰λ-lemma: ifW^u(p)has a topologically transverse intersection with W^s(q), thenW^u(p)C⁰-accumulates onW^u(q).

In [1] it is proved that if(0,0)∈int(ρ(ef))and f is aC¹⁺-diffeomorphism for some >0, then ef has a hyperbolic periodic saddle pointep∈R²such that

W^u(ep)tW^s(ep)+(a,b), (3) for all(a,b)∈Z²(W^u(ep)is the unstable manifold ofepandW^s(ep)is its stable manifold).

Note that, asep is a periodic point for ef, the same holds for all integer translations ofep and, moreover, for any integer vector(a,b),W^u,s(ep+(a,b))=W^u,s(ep)+(a,b).

In the area-preserving case, this result implies the following.

• W^u(ep)=W^s(ep)is a ef-invariant equivariant closed connected subset ofR²and there existsM=M(f) >0 such that any connected componentDeof(W^u(ep))^cis an open topological disk whose diameter is less thanM andD^def=^.π(eD)is a f-periodic disk.

Moreover, for any f-periodic diskD⊂T²,π⁻¹(D)⊂(W^u(ep))^c.

• For anyρ=(s/q,r/q)∈int(ρ(ef))∩Q², if we consider the map ef^q(•)−(s,r),then there exists a point ep_ρ that is a hyperbolic periodic saddle point for ef^q(•)−(s,r) whose stable and unstable manifolds have similar intersections to those in (3) and

W^u(ep_ρ)=W^s(ep_ρ)=W^u(ep)=W^s(ep).

So, the above set is the same for all rational vectors in the interior of the rotation set. We denote it byR.I.(ef)(region of instability of ef) and a similar definition can be considered in the torus: R.I.(f)^def=^.π(W^u(ep))=W^u(p), where p=π(ep)is f- periodic. Every f-periodic open disk inT²is contained in a connected component of the complement ofR.I.(f)and every such connected component is a f-periodic open disk, whose diameter when lifted to the plane is smaller thanM.

• Every open ball centered at a point of R.I.(f)has points with all rational rotation vectors contained in the interior ofρ(ef).

• If f is transitive, then ef is topologically mixing in the plane. This follows easily from the fact that if f is transitive, then R.I.(f)=T²and R.I.(ef)=W^u(ep)=W^s(ep)= R².

As we have already said, the above results were obtained in [1] under aC¹⁺condition.

In [13, 20], some analogous results were proved for homeomorphisms, by completely different methods, but the conclusions of some are weaker.

What about surfaces of higher genus?

In this setting, starting with the definition of rotation set, things are more involved. If Sis a closed orientable surface of genusg>1, the definition of rotation set needs to take into account the fact thatπ1(S),the fundamental group ofSandH₁(S,Z), the first integer homology group of S, are different: the first is almost a free group with 2g generators.

There is only one relation satisfied by the generators. While the second isZ^2g.

Possibly the most immediate consequence of this is the fact that in order to define a rotation set for surfaces of higher genus, if one does not want it to be too complicated but wants it to have some properties similar to what happens in the torus, a homological definition must be considered. In the following, we present the definition of a homological rotation set and a homological rotation vector as they appeared in [21]. The idea of using homology in order to define rotation vectors goes back to the work of Schwartzman [28].

1.2. Rotation vectors and rotation sets. LetS be a closed orientable surface of genus g>1 and letI: [0,1] ×S→Sbe an isotopy from the identity map to a homeomorphism

f :S→S.

For α a loop in S (a closed curve), [α] ∈H₁(S,Z)⊂H₁(S,R) is its homology class. Recall thatH₁(S,Z)'Z^2gand H₁(S,R)'R^2g.We will also consider H₁(S,R) endowed with the stable norm as in [12], which has the property thatk[γ]k ≤l(γ )for any rectifiable loopγ, wherel(γ )is the length of the loop.

the pathI_pinSfrompto f (p)andγfⁿ(p)traversed backwards, that is αⁿ_p=γp∗I_pⁿ∗γ⁻¹_fn(p).

We can now define the homological displacement function ofpas 9f(p)= [αp].

For the function9f :S→H1(S,R), we abbreviate its Birkhoff sums as 9ⁿ_f(p)=

n−1

X

k=0

9f(f^k(p)).

Note that, sinceαⁿ_pis homotopic toαp∗αf(p)∗ · · · ∗αfⁿ⁻¹(p), [αⁿp] =

n−1

X

k=0

[αf^k(p)] =

n−1

X

k=0

9f(f^k(p))=9ⁿ_f(p).

Also, the pathIⁿ_pcan be replaced by any path joining pto fⁿ(p)and homotopic with fixed endpoints toIⁿ_p. This implies that9f depends only on f,on the choice ofA_band on the homotopy class of the isotopyI.In particular,9f is bounded. Indeed, asSis compact, sup{d_D(eq, ef(eq)):

eq∈D} =Cmax f <∞, and if we replace the pathIpby the projection of the geodesic segment inDjoining ep∈π⁻¹(p)to ef(ep),as the length of this path is smaller thanC_{max f},thenk9fk ≤2C_Ab+C_{max f}.

As we just said, 9f depends on the choice of the basepoint b and the family Ab. However, given another basepointb⁰∈S and a family A⁰_b0 = {γp:p∈S}of rectifiable paths whose lengths are uniformly bounded byC_A⁰

b0 such thatγ_p⁰ joinsb⁰ to p, defining α⁰ⁿ_p analogously, one has

[α⁰ⁿ_p] = [γ_p⁰ ∗I_pⁿ∗γ⁰⁻¹_fn(p)] = [αⁿ_p∗δⁿ_p] = [δⁿ_p] +9ⁿ_f(p), (4) whereδⁿ_p=γfⁿ(p)∗γ⁰⁻¹_fn(p)∗γ_p⁰ ∗γ_p⁻¹. Indeed, the loopα⁰ⁿ_p is freely homotopic to Iⁿ_p∗ δⁿ_p. In particular, if9⁰_f(p)= [α⁰_p], then

k9ⁿf(p)−9⁰ⁿf (p)k ≤2CA_b +2CA0_b0. (5) Finally, if the limit

ρ(f,p)= lim

n→∞

1

n9ⁿ_f(p)∈H₁(S,R) (6) exists, we say thatphas a well-defined (homological) rotation vector.

After all this, we are ready to present the definition of the (homological) rotation set of f,which is analogous to the definition for the torus [24]. The Misiurewicz–Ziemian rotation set of f overSis defined as the setρmz(f)consisting of all limits of the form

v= lim

k→∞

1

n_k9ⁿ_f^k(pk)∈H1(S,R),

where pk∈S andnk→ ∞. By (5), the rotation set depends only on f,but not on the choice of the isotopy, the basepointbor the arcsγp. This definition coincides with

ρmz(f)= \

m≥0

[

n≥m

9ⁿ_f(p) n : p∈S

.

In particular, since9f is bounded, the rotation set is compact.

Note that, using a computation similar to (4), if one chooses a rectifiable arcβ joining fⁿ(p)top,

[I_pⁿ∗β] = [γ_p⁻¹∗αⁿ_p∗γfⁿ(p)∗β] =9ⁿ_f(p)+ [γfⁿ(p)∗β∗γ_p⁻¹]. (7) Thus,kI_pⁿ∗β−9ⁿ_f(p)k ≤2C_Ab +l(β).As a consequence, an alternate but equivalent definition of rotation vectors and rotation sets is obtained by considering all limits of the form

v= lim

k→∞

1 n_k[I_p^nk

k ∗βk],

where p_k∈S, n_k→ ∞ and βk are rectifiable arcs joining f^nk(p_k) to p_k such that l(βk) <∞.

Moreover, it is possible to choose the arcsγpin the definition of9f so that the map p7→9f is not only bounded, but also Borel measurable [11].

This is important if one wants to define rotation vectors of invariant measures. Let M(f)be the set of all f-invariant Borel probability measures. The rotation vector of the measureµ∈M(f)is defined as

ρm(f, µ)= Z

9f dµ∈H₁(S,R).

By the Birkhoff ergodic theorem, forµ-almost every point p∈S the limitρ(f,p)= lim_n→∞(1/n)9ⁿ_f(p)exists andρm(f, µ)=R ρ(f,p)dµ. Moreover, ifµis an ergodic measure, thenρ(f,p)=ρm(f, µ)forµ-almost every point p.

Due to these facts and (5), the rotation vector of a measure is also independent of any choices made in the definitions. Denote byρm(f)the rotation set of invariant measures, that is, ρm(f)=S

µ∈M(f)ρm(f, µ) and denote by ρerg(f) the corresponding set for ergodic measures. Then [24, proof of Theorem 2.4], without modifications, implies that

ρm(f)=Conv(ρerg(f))=Conv(ρmz(f)).

In particular, every extremal point of the convex hull ofρmz(f)is the rotation vector of some ergodic measure and, therefore, it is the rotation vector of some recurrent point.

The main problems with this definition of rotation set are the following.

• Although it is compact, it does not need to be convex.

1.3. A more precise motivation and statements of the main results. The main objective of this work is to give conditions which imply complicated and rich dynamics in the universal cover ofS, analogous to what happens for a homeomorphism of the torus isotopic to the identity when its rotation set contains(0,0)in its interior.

This type of problem has already been studied for surfaces of higher genus by Boyland in [4]. But, in that paper, he considered the Abelian cover ofS instead of the universal cover. As far as we know, this is the only published result on this kind of problem. Boyland considered homeomorphisms f :S→S of a special type, which are very important for our work: f is isotopic to the identity as a homeomorphism ofS, but it is pseudo-Anosov relative to a finite f-invariant setK⊂S(see [9]). He presented some conditions equivalent to f having a transitive lift to the Abelian cover ofS.

The hypotheses of our main results will imply, in particular, that if a C¹⁺

diffeomorphism f :S→S isotopic to the identity satisfies these hypotheses, then analogous results to those in [1] hold.

As a by-product of these results, we obtain that in theC¹⁺ setting, the homological rotation set is a compact convex subset ofR^2g which is 2g-dimensional: it is equal to the rotation set of the f-invariant Borel probability measures and all rational points in its interior are realized by periodic orbits. Non-rational points in the interior of the rotation set are also realized by compact f-invariant sets.

We are indebted to Alejandro Passeggi, who pointed out this consequence of Theorem 2 to us.

Moreover, as a corollary of the ideas used in this last result, we can extend the main theorems from [2] to our setting. This is done in Theorems 4 and 5.

In what follows, we precisely present the main results of this paper. Assume thatS is a closed orientable surface of genusg>1 andπ:eS→S is its universal covering map.

We may identify the universal covereSwith the Poincar´e diskDand denote by Deck(π) the groups of deck transformations ofS. Consider f :S→S, which is a homeomorphism isotopic to the identity, and letef :D→Dbe the endpoint of the lift of the isotopy from Id to f which starts at Id:D→D.We call ef the natural lift of f.

Definition 1.3. (Fully essential system of curves C) We say that f :S→S is a homeomorphism with a fully essential system of curvesC =Sk

i=1γi if the following conditions are satisfied:

(1) there exist different oriented closed geodesicsγ1, . . . , γk in S, k≥1, such that (Sk

i=1γi)^conly has non-essential connected components;

FIGURE1. An example where not all geodesics appear twice.

(2) for eachi∈ {1, . . . ,k},there is a f-periodic pointpi such that its trajectory under the isotopy is a closed curve freely homotopic toγi with the correct orientation;

(3) for every open intervalsI,F⊂∂D, there exists an oriented simple arceα⊂π⁻¹(C) formed by the concatenation of a finite number of oriented subarcs of extended lifts of geodesics inC and such that the initial point ofeαis contained inI and the final point belongs toF.

Remarks.

• No matter how largeg (the genus) is, it is possible to construct examples having a fully essential system of curves withk=2. Although the fundamental group has 2g generators, the number of geodesics may be much smaller.

• The third condition above is a little tricky to check. A much easier one, which implies it, is the following: for each i∈ {1, . . . ,k}, there are f-periodic points p⁻_i and p⁺_i such that their trajectories under the isotopy are closed curves freely homotopic toγi, or concatenations ofγi,with both possible orientations. In order to see that this implies the third condition, use Propositions 6 and 7. Nevertheless, we present this more general condition because what we really need about the fully essential system of curves is the property that, when considered as a connected subset of S, its complement only has disks as connected components and C contains oriented closed curves (the orientations are inherited by the orientations of theγi⁰s) whose homotopy classes generateπ1(S)as a semi-group. This is achieved, for instance, when C contains a generator forπ1(S),with each curve appearing twice, and with both possible orientations (as explained above), or, more generally, with any set of curves.

This more general situation is the one we describe in the third condition above. See the example in Figure 1, which shows a situation in which we can find generators for π1(S)as a semi-group in a fully essential system of curves but some curves do not appear twice with different orientations.

Now we present the main theorems in the order in which we prove them in the paper.

The exception is the first one, which we only sketch here, because its precise statement is more technical. The formal statement can be found in §3.

THEOREM 1. (Informal statement) Let f :S→S be a homeomorphism isotopic to the identity with a fully essential system of curvesC and let ef be its natural lift. Then there exists a real number c=c(f)≥0such that the ef -iterates of an open c-neighborhood of any fundamental domainQe⊂Dof S accumulate on all translates of the c-neighborhood ofQ under deck transformations and thus on the whole boundary ofe D.

to the finite invariant set of periodic points associated with the fully essential system of curvesC. Using several properties of the stable and unstable foliations of this map, it is possible to prove a result similar to Theorem 2 forφ, and then, using a theorem of Boyland [3] (see also [14]) and other technical results on Pesin theory [7,18], we can finally prove the theorem for the original map f.This procedure is similar to what was done in [1].

The main part of this paper is proving Theorem 2 for relative pseudo-Anosov maps and this is done in Lemma 13.

We would like to point out that the conclusion of Theorem 1 clearly implies the existence of a fully essential system of curves. In other words, Theorems 1 and 2 are both ‘if and only if’ statements.

The next results are consequences of Theorem 2, exactly as in [1]. They all share the same hypotheses: suppose that, for some >0, f :S→S is a C¹⁺ area-preserving diffeomorphism isotopic to the identity with a fully essential system of curvesC.

COROLLARY1. If f is transitive, then f cannot have a periodic open disk. In the general case, there exists M=M(f) >0such that if D⊂S is a f -periodic open disk, then for any connected componentD ofe π⁻¹(D),diam(eD) <M in the metric d_D,the lift of the hyperbolic metric d in S.

In [21], it is proved that in the case where f is just an area-preserving homeomorphism of S and the fixed point set is inessential, then all f-invariant open disks have diameter bounded by some constantM>0.If, moreover, for alln>0,the set ofn-periodic points is inessential, then, for eachn>0,the set ofn-periodic open disks has bounded diameter.

But the bound may not be uniform with the period. In our situation, with much stronger hypotheses, Corollary 1 gives a uniform bound.

COROLLARY2. There exists a contractible hyperbolic f -periodic saddle point p∈S (the one from Theorem 2) such that R.I.(f)^def=^.W^u(p)=W^s(p),is compact, f -invariant and all connected components of the complement of R.I.(f)are f -periodic disks. Moreover, for allep∈π⁻¹(p), R.I.(ef)^def=^.π⁻¹(R.I.(f))=W^s(ep)=W^u(ep)is a connected, closed,

ef -invariant, equivariant subset ofD.

COROLLARY3. If f is transitive, then there exists a contractible hyperbolic f -periodic saddle point p∈S (the one from Theorem 2) such that W^u(p)=W^s(p)=S and, for any ep∈π⁻¹(p), W^u(ep)=W^s(ep)=D,something that implies that ef is topologically mixing.

Finally, in the third theorem, we study the homological rotation setρmz(f).

THEOREM 3. Let f :S→S be a C¹⁺ diffeomorphism isotopic to the identity with a fully essential system of curvesC. Then the (homological) rotation setρmz(f)is a2g- dimensional compact convex subset of H1(S,R)'R^2g. Moreover, if v∈int(ρmz(f)), then there exists a compact set K⊂S such that, for all q∈K ,ρ(f,q)=v.In the case wherevis a rational point, K can be chosen as a periodic orbit.

The last two results generalize the main theorems of [2] to the context of this paper.

THEOREM4. Let f :S→S be a C¹⁺diffeomorphism isotopic to the identity with a fully essential system of curvesC. Then there exists M(f) >0such that, for anyω∈∂ρmz(f), any hyperplane ω∈H⊂R^2g that does not intersect interior(ρmz(f)) (H is called a supporting hyperplane), any p∈S and n>0,

([αⁿp] −n·ω)·−→

vH <M(f), where−→

vH is the unitary normal to H,which points towards the connected component of H^cthat does not intersectρmz(f).

THEOREM 5. Let f :S→S be a C¹⁺ area-preserving diffeomorphism isotopic to the identity with a fully essential system of curvesC. Then the rotation vector of Lebesgue measure belongs tointerior(ρmz(f)).

2. Some background, auxiliary results and their proofs

In this section, we present some important results that we will use, along with some definitions and a short digression on hyperbolic surfaces, Thurston classification of homeomorphisms of surfaces and a little of Pesin theory. We also prove some auxiliary results, which we will use in the following sections to prove Theorems 1–5.

2.1. Properties of hyperbolic surfaces. Let S be a closed orientable surface of genus g>1 and letπ:eS→S be its universal covering map. As we said before, the universal covereS is identified with the Poincar´e disk Dendowed with the hyperbolic metricd_D. Hence, we assume thatS=D/0, where0is a cocompact freely acting group of Moebius transformations. Any non-trivial deck transformation g∈Deck(π)=0 is a hyperbolic isometry and extends to the ‘boundary at infinity’∂D as a homeomorphism which has exactly two fixed points: one attractor and one repeller. These fixed points are the endpoints of some g-invariant geodesicδg of D, called the axis of g. For any point ep∈D, the sequence gⁿ(ep) converges to one endpoint of δg as n→ −∞and to the other one as n→ ∞. Any subarc ofδgjoining a point ep tog(ep),when projected to S,becomes an essential loopγg,which is the unique geodesic in its free homotopy class.

Given an essential loop γ : [0,1] →S, an extended lift of γ is an arc eγ:R→D obtained by the concatenation of arcs that are the translation of a lift ofγ by all iterates of some deck transformation. Two extended lifts of an essential loop coincide if and only if they share the same endpoints in∂D.

to a homeomorphism ofDas the identity on the ‘boundary at infinity’∂D(see [8]).

2.2. On the fully essential system of curves C. In this subsection, we prove some properties forπ⁻¹(C),whereC is a fully essential system of curves.

PROPOSITION6. The liftπ⁻¹(C)is a closed connected subset ofDthat accumulates all over∂Dwith the Euclidean metric.

Proof. First, observe that C is the union of a finite number of closed geodesics in S;

therefore C is closed. Since π:D→S is continuous, π⁻¹(C) is closed. To see that π⁻¹(C)is connected, we just observe thatS\C is a union of open topological disks, and therefore all connected components ofD\π⁻¹(C)are bounded topological open disks.

In order to prove that π⁻¹(C) accumulates everywhere in ∂D, we first note that, since, for allez∈Dand g∈Deck(π), π(ez)=π(g(ez)), π⁻¹(C)is invariant under deck transformations. This and the fact that the subset{

ez∈∂D:

ezis fixed by someg∈Deck(π)} is dense in∂Dwith the Euclidean metric (see [8]) imply that π⁻¹(C)accumulates all

over∂D.

In the next proposition, we consider the geodesics inC without their orientations.

PROPOSITION7. For everyep,er∈π⁻¹(C), there exists a pathγinπ⁻¹(C)joining these two points, which is contained in the union of a finite number of subarcs of extended lifts of geodesics inC.

Proof. Fix a point ep∈π⁻¹(C)and letP_epbe the set of all pointseq∈π⁻¹(C)such that there exists a path joiningeptoeqformed by subarcs of a finite number of extended lifts of geodesics inC. We will show thatP

epis an open and closed subset ofπ⁻¹(C). Leteq be a point inP

ep. As the setC is equal to the union of a finite number of closed geodesics, there exists >0 small enough so that B(eq)∩π⁻¹(C)satisfies one of the possibilities in Figure 2.

In the first case,eq belongs to just one extended lift of a geodesic inC. Ifγ is the path joiningeptoeq and it is formed byk>0 subarcs of extended lifts of geodesics, it is clear that, for all points inB(eq)∩π⁻¹(C),there is a pathγ⁰joiningepto this point formed by the same number of subarcs of extended lifts of geodesics. In the second case,eq belongs to the intersection of a finite number of extended lifts of geodesics and, again, if the path γ is formed byk>0 subarcs, then, for all points inB(eq)∩π⁻¹(C), there is a pathγ⁰

FIGURE2. Possibilities for a neighborhood ofeq.

joiningepto this point formed by at mostk+1 subarcs of extended lifts of geodesics. So Pepis open.

We will now prove thatP^c

ep=π⁻¹(C)\P

epis open. Again, ifeq is a point inP^c

ep,there exists >0 small enough such that B(eq)∩π⁻¹(C)satisfies one of the possibilities in Figure 2. In both cases, ifeq⁰∈B(eq)∩π⁻¹(C)∩P

ep, then, by the same argument as above, there is a pathγ⁰joiningeptoeqwith a finite number of subarcs of extended lifts of geodesics. But this is a contradiction becauseeq∈P^c

ep,so all points inB(eq)∩π⁻¹(C)are points ofP^c

ep. HenceP^c

epis open. SinceP

epis an open and closed subset of the connected setπ⁻¹(C),P

ep=π⁻¹(C).

2.3. Nielsen–Thurston classification of homeomorphisms of surfaces. In this subsection, we present a brief overview of Thurston’s classification of homeomorphisms of surfaces and prove a result analogous to [23, Theorem 1(i)].

2.3.1. Some definitions and the classification theorem. LetMbe a compact, connected, orientable surface, possibly with boundary, and let f :M→M be a homeomorphism.

There are two basic types of homeomorphisms which appear in the Nielsen–Thurston classification: the finite order homeomorphisms and the pseudo-Anosov ones.

A homeomorphism f is said to be of finite order if fⁿ=Id for some n∈N. The least suchnis called the order of f. Finite order homeomorphisms have zero topological entropy.

A homeomorphism f is said to be pseudo-Anosov if there is a real number λ >1 and a pair of transverse measured foliations F^u and F^s such that f(F^s)=λ⁻¹F^s and f(F^u)=λF^u. Pseudo-Anosov homeomorphisms are topologically transitive, have positive topological entropy and Markov partitions [9].

A homeomorphism f is said to be reducible by a system C=

n

[

i=1

C_i

of disjoint simple closed curvesC₁, . . . ,C_n, called reducing curves, if:

FIGURE3. Examples of a 1-prong and a 3-prong singularity, respectively.

• for alli,Ciis not homotopic to a point, nor to a component of∂M;

• for alli6= j,C_i is not homotopic toC_j; and

• Cis invariant under f.

THEOREM 8. (Nielsen–Thurston) If the Euler characteristic χ(M) <0, then every homeomorphism f :M→M is isotopic to a homeomorphismφ:M→M such that:

(1) φis of finite order;

(2) φis pseudo-Anosov; or

(3) φ is reducible by a system of curves C , and there exist disjoint open annular neighborhoods Ui of Ci such that

U=[

i

Ui

isφ-invariant. Each component S_iof M\U is mapped to itself by some least positive iterate ni ofφ, and eachφⁿⁱ|_S

i satisfies (1) or (2). Each U_i is mapped to itself by some least positive iterate m_iofφfixing the boundary components, and eachφ^mi|_U

i

is a generalized twist.

Homeomorphismsφas in Theorem 8 are called Thurston canonical forms for f. We say that φ:M→M is pseudo-Anosov relative to a finite invariant set K if it satisfies all of the properties of a pseudo-Anosov homeomorphism except that the associated stable and unstable foliations may have 1-pronged singularities at points in K [15], see Figure 3. Equivalently, letN be the compact surface obtained from M\K by compactifying each puncture with a boundary circle and let p:N→M be the map that collapses these boundary circles to points. Thenφis pseudo-Anosov relative to K if and only if there is a pseudo-Anosov homeomorphism8:N→N such thatφ◦p=p◦8.

2.3.2. The beginning of the work. The following result is the first step towards the proof of the main theorems.

LEMMA 9. Let f :S→S be a homeomorphism isotopic to the identity with a fully essential system of curvesC and let P be the set of periodic points associated with the geodesics inC. Then there exists an integer m0>0such that f^m0is isotopic relative to P to a homeomorphismφ:S→S, which is pseudo-Anosov relative to P.

Proof. Let f be a homeomorphism with a fully essential system of curves C and let P be the set of all periodic points associated with the geodesics in C. We write P= {p₁,p₂, . . . ,p_k}. For each 1≤i≤k, there exists an integer n_i>0 such that fⁿⁱ(p_i)=p_i. Take m0>0 to be an integer such that all points in P are fixed points for f^m0.

We will follow the same ideas used by Llibre and MacKay in [23]. Letφ:S→Sbe the Thurston canonical form associated to f^m0.Of course, we are considering f^m0:S\P→ S\P and soφis also a homeomorphism from S\P into itself. But it can be extended in a standard way to the set P(fixing everybody), giving a homeomorphism ofS into itself that is also isotopic to the identity as a homeomorphism ofS,which we still callφ.

Let us show thatφis pseudo-Anosov relative toP.First, note thatφcan not be of finite order, since points in π⁻¹(P)move in non-trivial homotopical directions. To be more precise, ifφhad finite order, then, for someN>0,φ^N≡Id.This implies that the natural lift ofφ^N is also the identity. But there is at least one fixed point forφ,p₁∈P,such that, for anyep1∈π⁻¹(p1),its trajectory under the natural lifteφ^N:D→Dfollows a non-trivial deck transformation.

Now, suppose φ is reducible by a system of curves C. As in [23], we say that a simple closed curveγ on a surface of genusg with punctures is non-rotational if, after closing the punctures,γ is homotopically trivial. Ifγ is a non-rotational reducing curve, then it must surround at least two punctures. So, supposeγ surrounds pi and pj,i6= j. Since γ is a reducing curve, φⁿ(γ )=γ, for somen>0. This means that there exists g∈Deck(π)such thateφⁿ(eγ )=g(eγ ), whereeγ is a lift ofγ (eγ is a simple closed curve inD)surroundingep_i andep_j, which are lifts of p_i and p_j, respectively. By induction, it follows thateφ^mn(eγ )=g^m(eγ )encloses botheφ^mn(ep_i)andeφ^mn(ep_j)for allm∈Z. But this is a contradiction because asi6= j, liml→∞eφ^l(epi)and liml→∞eφ^l(epj)are different points of∂D.

In the case where γ is a rotational reducing curve, leteγ⊂D be an extended lift of γ. The curveeγ has two distinct endpoints at the ‘boundary at infinity’∂D,andD\

eγ has exactly two connected components. Sinceeφ|_∂D=Id,eφ(eγ )has the same endpoints on∂D aseγ. SinceS\C is a union of topological disks, there existsg∈Deck(π)associated with some geodesicγiinC such that the fixed points ofgin∂Dseparate the endpoints ofeγ.

Finally, chooseep_i ∈π⁻¹(p_i)such that it belongs to one connected component ofD\ eγ and lim_n→∞eφⁿ(ep_i)is in the ‘boundary at infinity’ of the other connected component.

Sinceeφ(eγ )andeγhave the same endpoints in∂Dandφⁿ(γ )=γ,we haveeφ^mn(eγ )= eγ, for allm>0. Aseφpreserves orientation, this clearly implies a contradiction (see Figure 4).

This shows thatφcannot be of finite order or reducible by a system of curves. So φis

pseudo-Anosov relative toP.

2.4. On Handel’s fixed point theorem.

2.4.1. Preliminaries and a statement of Handel’s theorem. In [16], Michael Handel proved the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk as the identity on the boundary, provided that, for certain points in the open disk, theirαandω-limit sets are single points

FIGURE4. The final contradiction.

in the boundary of the disk, distributed with a certain cyclic order. Later, in [22], Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments. In Le Calvez’s proof, the existence of the fixed point follows from the existence a simple closed curve contained in the open disk, whose topological index can be calculated and is equal to one.

THEOREM10. (Handel’s fixed point theorem, [22]) Consider a homeomorphismeh:D→ Dof the closed unit disk satisfying the following hypotheses.

(1) There exists r≥3 points ep₁, . . . ,ep_r in D and 2r pairwise distinct points α1, ω1, . . . , αr, ωr on the boundary∂Dsuch that, for every1≤i≤r ,

n→∞lim eh⁻ⁿ(ep_i)=αi, lim

n→∞ehⁿ(ep_i)=ωi. (2) The cyclic order on∂Dis, as represented on Figure 5,

α1, ωr, α2, ω1, α3, ω2, . . . , αr, ωr−1, α1.

Then there exists a fixed point free simple closed curveγ ⊂Dsuch thatind(eh, γ )=1.

Remember that, ifepis an isolated fixed point ofeh, the Poincar´e–Lefschetz index ofeh atepis defined as

ind(eh,ep)=ind(eh, γ ),

whereγis a (small) simple closed curve surroundingepand no other fixed point. The index ofehatepdoes not depend of the choice ofγ.

In the case whereeh has only isolated fixed points, if int(γ )is the bounded connected component of γ^c and Fix(int(γ ))= {

ep∈int(γ ):eh(ep)=

ep} then, by properties of the Poincar´e–Lefschetz index,

ind(eh, γ )= X

ep∈Fix(int(γ ))

ind(eh,ep).

So, ifeh:D→Dis a homeomorphism with only isolated fixed points satisfying the hypothesis of Handel’s theorem, as ind(eh, γ )=1,there exists a fixed point ep⁰∈int(γ ) with ind(eh,ep⁰) >0.

FIGURE5. Cyclic order for Handel’s fixed point theorem whenr=3 andr=5.

2.4.2. Existence of a hyperbolic φe-periodic point. Remember that φ:S→S is a homeomorphism which is pseudo-Anosov relative to P (see Lemma 9). As a map from S to itself, φ is a homeomorphism isotopic to the identity. The map eφ:D→Dis the natural lift ofφ,the one which commutes with all deck transformations and extends as a homeomorphism ofD,which is the identity on the ‘boundary at infinity’.

In the next proposition, we prove thateφhas a hyperbolic periodic saddle point. When we say hyperbolic saddle in this context, we mean that the local dynamics at the point is obtained by gluing exactly four hyperbolic sectors, or, equivalently, the point is a regular point of the foliationsF^uandF^s.

PROPOSITION 11. The natural lift eφ:D→D of the map φ from Lemma 9 has a hyperbolic periodic (saddle) pointep.

Proof. In the first part of this proof, we want to find a well-oriented Jordan curve βe contained inπ⁻¹(C).After finding such a curve, we consider interior(eβ)∩π⁻¹(C)^c.We will show that there is a connected componentUeof the previous open set, whose boundary is also a well-oriented Jordan curve. Finally, taking appropriate lifts of the periodic points associated with the geodesics inC which have extended lifts containing arcs in∂Ue,we get that the hypotheses of Handel’s theorem are satisfied for them.

First, choose some unoriented geodesicsα1, α2, . . . , αr,for somer≤k,such that, as a set,Sr

i=1αi=Sk

i=1γi,whereC =Sk

i=1γi is the fully essential system of curves.

If, for every 1≤i≤r,there are two periodic points whose trajectories under the isotopy are closed curves freely homotopic to αi, or concatenations of αi, with both possible orientations, then any Jordan curve which is the boundary of a connected component of π⁻¹(C)^ccan be well oriented according to the orientations of the geodesics inC.This is what we need.

So, assume that the above does not hold and choose some oriented extended lifteγaof a geodesicγa inC, for which there is no periodic point following it with the opposite orientation.

FIGURE6. How to find the well-oriented Jordan curveβ.e

Letgabe a deck transformation which haseγaas axis and translates points according to the orientation ofeγa.Denote the two connected components ofeγa^cas Oe⁺ andOe⁻.It is clearly possible to choose two oriented geodesics inC,such that, for some extended lifts of them, one starts in∂Oe⁻\

eγaand ends in∂Oe⁺\

eγaand the other one goes in the opposite direction. Denote these lifts byeγb andeγc.Iterating them underg_a, if necessary, we can suppose that they are disjoint and their relative position is as in Figure 6.

Still considering Figure 6, leteαbe an oriented simple arc contained inπ⁻¹(C)which starts at some point in the open intervalI⊂∂Dand ends at some point in the other open intervalF⊂∂D.Remember that, asC is a fully essential system of curves, it is possible to choose such an arceαformed by the concatenation of finitely many oriented subarcs of extended lifts of geodesics inπ⁻¹(C).

Ifeα∩

eγahas two or more points, then, clearly,eα∪

eγa contains a well-oriented Jordan curveβ.eThis follows from the choice ofγa:there is no periodic point inSwith a lift that followseγain the opposite orientation.

If not, then it is still easy to find a well-oriented Jordan curveeβcontained ineα∪ eγa∪ eγb∪

eγc.

Now, let us look at interior(eβ).If π⁻¹(C)intersects interior(eβ), pick any extended lifteη⊂π⁻¹(C) that intersects interior(β).e The oriented arceη divides interior(eβ) into finitely many disks, at least one of them with a well-oriented boundary, still contained in π⁻¹(C).Denote this boundary byβe1,which, as we just said, is a well-oriented Jordan curve contained in π⁻¹(C). If π⁻¹(C)intersects interior(eβ1), repeat the process and find a well-oriented Jordan curveβe2⊂π⁻¹(C), and so on. As there are only finitely many extended lifts of geodesics inπ⁻¹(C)that intersect interior(β),e after finitely many steps, we arrive at a well-oriented Jordan curveβe∗⊂π⁻¹(C)such thatπ⁻¹(C)does not intersectUe=interior(βe∗).

Fix an oriented side eρ of ∂Ue=βe∗, which is given by the intersection of a certain extended lift of a geodesicγi1 inC withβe∗.Denote this extended lift byeγi1.Associated

FIGURE7.Ueand how some points move with respect to its boundary.

witheγi1,we can find an appropriate liftep_i1ofp_i1followingeγi1 with the correct orientation under iterates ofeφ.

Ifeρandρe⁰are two consecutive oriented sides of∂Ue, then the endpoints of the extended lift of the geodesic associated to eρ separate the endpoints of the extended lift of the geodesic associated toeρ⁰. Putting all these observations together, we see thateφsatisfies the hypotheses of Handel’s theorem (see Figure 7).

Sinceφis pseudo-Anosov relative to a finite setP,for each period, it has only isolated periodic points, and the same holds foreφ. This means, by Handel’s theorem, that there exists a fixed pointep₁ofeφsuch that

ind(eφ,ep₁)=ind(φ, π(ep₁)) >0. Observe that the same conclusion holds forφe^m, for anym>0.

But, for some appropriate largem₁>0,the local dynamics at points in Fix(φ)imply that

ind(φ^m1,p)≤0 for all p∈Fix(φ).

This happens because all points in Fix(φ) with non-positive indexes are saddle-like (maybe with more than four sectors) with φ-invariant separatrices, and points with positive indexes are rotating saddles. So, for some m1>0 sufficiently large, φ^m1 fixes the separatrices of all points in Fix(φ), and thus they all have non-positive indexes with respect toφ^m1.In particular, ind(φ^m1, π(ep1)) <0.

Now let us look atφ^m1.Again, as a consequence of Handel’s theorem, there is a fixed pointep2ofeφ^m1 with ind(eφ^m1,ep2)=ind(φ^m1, π(ep2)) >0.In the same way as above for some sufficiently largem₂>0,the local dynamics at points in Fix(φ^m1)imply that

ind(φ^m1m2,p)≤0 for allp∈Fix(φ^m1), and, in particular, ind(φ^m1m2, π(ep₂)) <0.

If we continue this process, we get a sequence of pairwise different points ep₁,ep₂,ep₃, . . .. InS,the pointsπ(ep₁), π(ep₂), π(ep₃), . . .are also pairwise different.

state some definitions and properties of pseudo-Anosov maps relative to finite invariant sets, which will be useful in the proof of the next lemma.

Let p∈S be a fixed point of φ. As we already said, the dynamics of a sufficiently large iterate ofφin a neighborhood ofpcan be obtained by gluing finitely many invariant hyperbolic sectors together. In each sector, the dynamics are locally like the dynamics in the first quadrant of the map(x,y)7→(λ1x, λ2y), for some real numbers 0< λ2<1< λ1. We define the stable set ofpas the setW^s(p)of pointszinSsuch thatφⁿ(z)→ pwhen n→ ∞, and we define the unstable set of p as the setW^u(p)of pointszinS such that φ⁻ⁿ(z)→p whenn→ ∞. If pis a regular point of the foliationsF^s,F^u, thenW^u(p) is the union of two branches; the same is true forW^s(p).This is the situation in which we called the point a hyperbolic saddle point in the previous proposition. In the case wherep is a singular point of the foliations, pis ak-prong singularity (fork=1 or somek≥3), which implies thatW^u(p)is the union ofkbranches; the same is true forW^s(p).In this singular case, each branch is actually a leaf of the proper foliation, which emanates from the singularity, while, in the regular case, each leaf gives two branches. In both the regular and the singular cases, the branches are either invariant or rotated aroundpunder iterates ofφ(and are thusφⁿ-invariant for somen>0).

In the case wherep⁰∈Sis aφ-periodic point, ifn_p⁰ is the least period of p⁰, then it is a fixed point ofφ^np0, so we define the stable and unstable sets of p⁰accordingly, usingφ^np0 instead ofφ.

LEMMA12. Leteφbe the natural lift ofφ. Then there existsep∈Daeφ-hyperbolic periodic saddle point and deck transformations g1, g2such that g1◦g26=g2◦g1and

Fe^u+

ep tFe_g^s+

i(ep), i∈ {1,2}, where W^u(ep)=eF^u+

ep ∪eF^u−

ep , W^s(ep)=Fe^s+

ep ∪Fe^s−

ep and Fe^u+

ep ,eF^u−

ep ,Fe^s+

ep ,eF^s−

ep are the four branches atep.

Proof. Let ep∈D be theeφ-periodic point given in Proposition 11. So, p=π(ep)is a hyperbolicφ-periodic saddle point. Without loss of generality, considering an iterate of φ, if necessary, we will assume that each point in K = {p} ∪P is fixed and, moreover, that each stable or unstable branch at a point inK is also invariant underφ.

The map φ is pseudo-Anosov relative to P. In particular, any stable leaf F^s∈F^s intersects all unstable leaves F^u∈F^u C¹-transversely and vice-versa. Let F_p^u be the unstable leaf at the point p (as p is regular, F^u_p=W^u(p)) and let F_∗^s_p0 be a stable leaf at some pointp⁰∈P= {p₁, . . . ,p_k}.The point p⁰may be singular or regular. From what

we said above, F_p^utF_∗p^s 0. So, there exists an unstable branch at p, denoted by F^u+_p , and an unstable branch at p⁰,denoted by F_∗p^u00,such thatF_p^u+ accumulates onF_∗p^u00 and F_∗p^u00tW^s(p).LetF_p^s+be a stable branch at p such thatF_∗p^u00tF_p^s+.Lifting everything to the universal cover, having fixed some ep∈π⁻¹(p),there exist deck transformations g⁰6=Id andhsuch that

eF^u+

ep teF₍^s+_g0)ⁿh(ep), (8) for all sufficiently largen>0. This follows from the fact that, having fixed some ep∈ π⁻¹(p),there exist a ep⁰∈π⁻¹(p⁰)and deck transformations g⁰6=Id and h such that eφ(ep⁰)=g⁰(ep⁰),Fe^u+

ep tFe_∗^s

ep⁰ andFe_∗^u0

ep⁰tFe_h^s+_(ep).

Letg₁=(g⁰)ⁿh for somen>0 such that (8) holds. Now considereθto be a path inD constructed as follows:eθ=eθ⁰∗eθ⁰⁰, whereeθ⁰is a compact subarc ofFe^u+

ep starting atepand ending at a point in eF^u+

ep ∩eF_g^s+

1(ep),andeθ⁰⁰ is a compact subarc of eF_g^s+

1(ep) starting at the endpoint ofeθ⁰and ending atg1(ep).

Letω1be the fixed point in∂Dofg₁such that lim_n→∞gⁿ₁(eq)=ω1for alleq∈D, and letα1be the other fixed point.

Define

2=[

i∈Z

g₁ⁱ(eθ).

By construction, 2is a path connected subset ofD,joining α1 toω1. Since S\C is a union of open topological disks, there exists an oriented geodesic γ^∗ inC andm∈ Deck(π)such that the projection of the oriented axis ofminSisγ^∗and the fixed points ofmin∂Dseparate the endpointsω1andα1of2. This follows from Propositions 6 and 7.

Now consider the fixed pointsωm andαm of m in ∂D such that limn→∞mⁿ(eq)= ωm and lim_n→−∞mⁿ(eq)=αm, for alleq∈D.We know that the axis ofmis an oriented extended lift of γ^∗,so ωm andαm are coherent with the orientation of γ^∗. Letn0>0 be a sufficiently large integer such thatmⁿ⁰(ω1)andmⁿ⁰(α1)are close to ωm and2∩ mⁿ⁰(2)= ∅. This is possible because2accumulates onωm under positive iterates ofm.

Then

2=[

i∈Z

gⁱ₁(eθ)⇒mⁿ⁰(2)=[

i∈Z

mⁿ⁰gⁱ₁(eθ).

As eφ commutes with all deck transformations, eθ⁰⊂eF^u+

ep and eφ(Fe^u+

ep )=eF^u+

ep , we get that, for alln>0 and t∈Deck(π), t(eθ⁰)⊂eφⁿ(t(eθ⁰)). Similarly, sinceeθ⁰⁰⊂eF_g^s+

1(ep), eφⁿ(t(eθ⁰⁰))⊂t(eθ⁰⁰), for alln>0 andt∈Deck(π).

The hypotheses on C imply that there is a point ep_m∈π⁻¹(P) such that eφ(ep_m)= m(ep_m)andep_m is in the connected component ofD\2that containsαm in its boundary.

As mⁿ⁰(2) is in the other connected component of D\2, lim_n→∞φeⁿ(ep_m)=ωm and eφ|_∂

D=Id,we get that, for a sufficiently largen⁰>0, there must exists two integersi⁰, i⁰⁰such that

eφⁿ⁰(g₁ⁱ⁰(eθ⁰))tmⁿ⁰gⁱ₁⁰⁰(eθ⁰⁰).

In particular,Fe^u+

g₁ⁱ⁰(ep)tFe^s+

mⁿ⁰g₁ⁱ⁰⁰(ep), and so,Fe^u+

ep tFe^s+

(g⁻ⁱ₁ ⁰mⁿ0gⁱ₁⁰⁰)(ep)(see Figure 8).

Finally, letg2=g₁⁻ⁱ⁰mⁿ⁰g₁ⁱ⁰⁰. We will show thatg1andg2do not commute. Ifg1◦g2= g₂◦g₁,then there existsl∈Deck(π)and integersk₁,k₂such thatg₁=l^k1 andg₂=l^k2.

FIGURE8. How to obtaing₂.

Thus

g⁻ⁱ₁ ⁰mⁿ⁰gⁱ₁⁰⁰=l^k2 ⇒l^−i0k1mⁿ⁰l^i00k1=l^k2 ⇒mⁿ⁰ =l^{k2+k1(i0−i00)}.

Sincemⁿ⁰ andg1are iterates of the same deck transformation, the geodesics associated to the axes ofmandg₁are equal. But this is in contradiction with our choice ofm. So,g₁

andg2do not commute.

2.6. Proof of Theorem 2 in a special case. In this subsection, we prove Theorem 2 in case of relative pseudo-Anosov maps.

Remark 2.6. Asφ:S→Sis pseudo-Anosov relative to a finite invariant set, if, for some leaves F^u ofF^u and F^s ofF^s,there are connected components Fe^u of π⁻¹(F^u)and eF^s of π⁻¹(F^s)which have non-empty intersection (not at a lift of a singularity of the foliations), then they intersect in aC¹-transverse way. In the proof of the next lemma, we will not make use of this fact because, when proving Theorem 2, at some point we say that the proof continues as the proof of the next lemma. So, in the proof of Lemma 13, although intersections between stable and unstable leaves, either inSor inD,are always C¹-transverse, we will not use this fact.

Moreover, as we said in the introduction, the main feature of topologically transverse intersections is the fact that aC⁰-version of the so calledλ-lemma (see [26]) holds: if M is a surface, f :M→M is aC¹ diffeomorphism, p,q∈M are f-periodic saddle points andW^u(p)has a topologically transverse intersection with W^s(q),then W^u(p) C⁰-accumulates onW^u(q),and, in particular,W^u(p)⊃W^u(q).So if p1,p2,p3∈M are hyperbolic f-periodic saddle points, W^u(p₁)has a topologically transverse intersection withW^s(p₂)andW^u(p₂)has a topologically transverse intersection withW^s(p₃),then W^u(p1)has a topologically transverse intersection withW^s(p3).

LEMMA 13. (Theorem 2 in case of relative pseudo-Anosov maps) Let eφbe the natural lift of the mapφ. Then there exists a contractible hyperbolicφ-periodic point p∈S,such